Product expansions of q-character polynomials

• Published in: Journal of algebraic combinatorics Vol. 57; no. 3; pp. 975 - 1005
• Main Authors: Balachandran, Adithya, Gadish, Nir, Huang, Andrew, Sun, Siwen
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.05.2023; Springer Nature B.V
• Subjects: Combinatorics; Computer Science; Convex and Discrete Geometry; Group Theory and Generalizations; Lattices; Mathematical analysis; Mathematics; Mathematics and Statistics; Order; Ordered Algebraic Structures; Polynomials; Rings (mathematics); Statistics; Tensors; Matrix statistics; 05E05; Character polynomials; 20G40; 05A10; Finite general linear group
• ISSN: 0925-9899; 1572-9192
• DOI: 10.1007/s10801-022-01208-0

The ring of q -character polynomials is a q -analog of the classical ring of character polynomials for the symmetric groups. This ring consists of certain class functions defined simultaneously on the groups Gl n ( F q ) for all n , which we also interpret as statistics on matrices. Here, we evaluate these statistics on all matrices and work toward computing the structure constants of the product in this ring. We show that the statistics are periodically polynomial in q and governed by universal polynomials P λ , μ ( q ) which we compute explicitly, indexed by pairs of integer partitions. The product structure is similarly polynomial in q in many cases, governed by polynomials R λ , μ ν ( q ) indexed by triples of partitions, which we compute in some cases. Our calculations seem to exhibit several unexpected patterns. Mainly, we conjecture that certain indecomposable statistics generate the whole ring and indeed prove this for statistics associated with matrices consisting of up to 2 Jordan blocks. Furthermore, the coefficients we compute exhibit surprising stability phenomena, which in turn reflect stabilizations of joint moments as well as multiplicities in the irreducible decomposition of tensor products of representations of Gl n ( F q ) for n ≫ 1 . We use this stabilization to compute the correlation of the number of unipotent Jordan blocks of two sizes.